Divisorial persistent homology and volume asymptotics of complex pairs
Abstract
To a log-resolution of a complex pair (X,I) we associate the weights λE=(aE+1)/νE of the prime divisors E of the total transform of I, where νE is the vanishing order of I and aE the Jacobian (Mather discrepancy) order along E. On Jacobian-adapted resolutions, minEλE is the volume-growth exponent of KI=∑j|fj|2 for the 2n-dimensional Hausdorff measure: the log canonical threshold for smooth X, Ishii's Mather one for singular X, with a logarithmic correction given by a resonance multiplicity. The main object is the divisorial persistent homology DPH: the persistent homology of the dual complex filtered by the subcomplexes on divisors of weight at most α. It records incidences of divisors, which the threshold ignores. We prove that DPH is invariant under blow-ups along smooth centres in normal crossing with the boundary, for adapted resolutions; A2 shows adaptedness is necessary. Independence of the resolution holds for complex algebraic pairs (weak factorization) and normal surface singularities, and is a hypothesis for non-algebraic analytic pairs of dimension ≥3. When V(I) is complete, DPH equals the weight-zero part of the cohomology of the union of divisors of weight at most α (Deligne). For An the minimal resolution is adapted iff n is odd, and DPH0 has one finite bar of length (n−1)/2, so within this family it detects n (detection, not classification). We compare with Nicaise--Xu and de Fernex--Kollár--Xu.
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BibTeX
@misc{grulha2026divisorial,
title = {{Divisorial persistent homology and volume asymptotics of complex pairs}},
author = {Nivaldo Grulha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.06717},
eprint = {2607.06717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.06717},
}
RIS
TY - UNPB
TI - Divisorial persistent homology and volume asymptotics of complex pairs
AU - Nivaldo Grulha
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2607.06717
UR - https://arxiv.org/abs/2607.06717
AB - To a log-resolution of a complex pair (X,I) we associate the weights λ_E=(a_E+1)/ν_E of the prime divisors E of the total transform of I, where ν_E is the vanishing order of I and a_E the Jacobian (Mather discrepancy) order along E. On Jacobian-adapted resolutions, min_Eλ_E is the volume-growth exponent of K_I=∑_j|f_j|^2 for the 2n-dimensional Hausdorff measure: the log canonical threshold for smooth X, Ishii's Mather one for singular X, with a logarithmic correction given by a resonance multiplicity. The main object is the divisorial persistent homology DPH: the persistent homology of the dual complex filtered by the subcomplexes on divisors of weight at most α. It records incidences of divisors, which the threshold ignores. We prove that DPH is invariant under blow-ups along smooth centres in normal crossing with the boundary, for adapted resolutions; A_2 shows adaptedness is necessary. Independence of the resolution holds for complex algebraic pairs (weak factorization) and normal surface singularities, and is a hypothesis for non-algebraic analytic pairs of dimension ≥3. When V(I) is complete, DPH equals the weight-zero part of the cohomology of the union of divisors of weight at most α (Deligne). For A_n the minimal resolution is adapted iff n is odd, and DPH_0 has one finite bar of length (n−1)/2, so within this family it detects n (detection, not classification). We compare with Nicaise--Xu and de Fernex--Kollár--Xu.
ER -